Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

Finite nondegeneracy is interesting for the following reason. In the sequel,we shall have to consider an infinite collection of equations of the form(5.7) Θ j,β (t) + ∑γ∈N m ∗γ (β + γ)!(ζ) Θ j,β+γ (t) = ω j,β ,β! γ!where N m ∗ := N m \{0}, where j runs from 1 to d, where β runs in N mand where the right hand sides ω j,β are independent complex variables. Forβ = 0, the equations (5.7) write simply Θ j (ζ, t) = ω j,0 . By definition,if M is l 0 -nondegenerate at 0, there exists n integers j∗, 1 . . .,j∗n with 1 ≤j∗ i ≤ d and n multi-indices β1 ∗ , . . ., βn ∗ ∈ N m with |β∗ i| ≤ l 0 such thatthe local holomorphic self-mapping t ↦→ (Θ j k ∗ ,β (t)) ∗ k 1≤k≤n of C n is of rankn at the origin. Considering the equations (5.7) for j = j∗ 1, . . .,jn ∗ andβ = β∗, 1 . . ., β∗ n and applying the implicit function theorem, we observe thatwe can solve t in terms of (ζ, ω j 1 ∗ ,β∗ 1, . . ., ω j∗ n ,β∗ n ) by means of a holomorphicmapping, namely(5.8) t = Ψ(τ, ω j 1 ∗ ,β 1 ∗ , . . .,ω j n ∗ ,βn ∗ ).Without loss of generality, we may assume that Ψ is holomorphic for |ζ|